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ODE question

  1. Jun 24, 2008 #1
    1. The problem statement, all variables and given/known data
    given this ODE with initial conditions y(1)=0
    [tex] \[
    (x + y^2 )dx - 2xydy = 0
    2. Relevant equations
    solving this ODE gives us
    [tex]\[y = \sqrt {x\ln (x)} \][/tex]
    as we can see this equation is true only for x>=1
    in order to use the theorem on existence and uniqueness we isulate for y'=f(x,y)
    [tex]\[y' = \frac{{(x + y^2 )}}{{2xy}}\][/tex]
    and we can see that when y=0 the equation is not defined
    3. The attempt at a solution
    my question is
    1) if x>=1 does that mean that the bound for y is y>=0?
    2)if it meaas that y>=0 then should i conclude that the theorem on existence and uniqueness does not apply here since the function is not continuous thus we cant say that the solution is unique? what does it mean
    thanks for the help
  2. jcsd
  3. Jun 25, 2008 #2


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    Staff Emeritus
    Science Advisor

    Yes, since [itex](x+y^2)/(2xy)[/itex] does not exist when y=0, the "existence and uniqueness" theorem does not hold here. That means we cannot conclude that a unique solution exists on any open interval around x= 1. But it might be true anyway.

    What you have done is find a solution, [itex]y= \sqrt{xln(x)}[/itex], That is defined only for x> 1 (and y must be positive)- that is not defined on an open interval containing x= 1. But, in fact, what happens here is that there DOES exist a solution on an interval around x= 1: y= 0 for all x is such a second solution. The solution is not unique on the x> 1.
  4. Jun 25, 2008 #3

    if can please elaborate more i dont think i understand what you mean
  5. Jun 25, 2008 #4


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    Staff Emeritus
    Science Advisor

    What part do you not understand? Surely it is obvious that y= 0 for all x is a solution to that problem.
  6. Jun 26, 2008 #5
    well y=0 is not a solution as i see it if i put it in my original equation it doesnt solve it

    what i dont understand is if for these initial conditions the theorem on uniqueness and existence apply or not? and if it doesnt apply what can i say on the solution i found hence is it a solution or not?
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